The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA

Yuanyuan Zhang; Zhaofeng Li

doi:doi:10.11648/j.acm.20180703.22

| Peer-Reviewed

The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA

Yuanyuan Zhang, Zhaofeng Li

Published in Applied and Computational Mathematics (Volume 7, Issue 3)

Received: 9 August 2018 Published: 13 August 2018

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In 1974, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing. In 1986, Y. Meryer created a real small wave base, and the wavelet analysis began to flourish after a multi scale analysis of the same method of constructing the small wave base with S. Mallat. In order to analyze and deal with non-stationary signals, a series of new signal analysis theories are proposed.: Short Time Fourier Transform, time-frequency analysis, wavelet transform, and fractional Fourier transform and so on. In this paper, an explicit algorithm is given to construct the minimum-energy frames based on frame multiresolution analysis via characteristic vectors of the mask matrix. In section 2, we show the structure of minimum-energy wavelet frames in terms of their masks (Lemma 1) and discuss that we should eliminate the correlation of the rows of the mask matrix by the polyphase decomposition technique. Based on FMRA, an explicit algorithm is given to construct this frames. By this method, all the minimum-energy wavelet frames can be obtained. As an application, several examples are showed to explain this method in section 3. This method can also be applied in other fields of wavelet analysis.

Published in	Applied and Computational Mathematics (Volume 7, Issue 3)
DOI	10.11648/j.acm.20180703.22
Page(s)	161-166
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Frame Multiresolution Analysis, Polyphase Decomposition, Minimum-Energy Frames

References

[1]	S. Mallat, “Multiresolution approximations and orthonormal bases of wavelets for L2 (R),” Trans. Amer. Math. Soc., vol. 315, 1989, pp. 69-87.
[2]	P. Steffen, P. N. Heller, “R. A. Gopinath and C. S. Burrus, Theory of regular M-band wavelet bases,” IEEE Trans. Signal Processing, vol. 41 (1993), 3497-3510.
[3]	C. Chaux and L. Duval, “Image analysis using a dual-tree M-band wavelet transform,” IEEE Transactions on Image Processing, vol. 15, 2006, pp. 2397-2412.
[4]	M. K. Mihcak, I. Kozintsev, K. Ramchandran and P. Moulin, “Low-complexity image denoising based on statistical modeling of wavelet coefficients,” IEEE Signal Processing Letters, vol. 6, 1999, pp. 300- 303.
[5]	L. Gan and K. K. Ma, A simplified lattice factorization for linear-phase perfect reconstruction filter bank, IEEE Signal Processing Letters, vol. 8, 2001, pp. 207-209.
[6]	B. Han, Symmetric orthonormal scaling functions and wavelets with dilation factor 4, Adv. Comput. Math. Vol. 8, 1998, pp. 221-247.
[7]	Chao, Zhang, et al. "Optimal scale of crop classification using unmanned aerial vehicle remote sensing imagery based on wavelet packet transform." Transactions of the Chinese Society of Agricultural Engineering (2016).
[8]	Shleymovich M. P., M. V. Medvedev, and S. A. Lyasheva. "Object detection in the images in industrial process control systems based on salient points of wavelet transform analysis." International Conference on Industrial Engineering, Applications and Manufacturing IEEE, (2017):1-6.
[9]	I. Daubechies, “Ten lectures on wavelets,” CBMF conference series in applied mathematics 61, SIAM, Philadelphia, 1992.
[10]	J. J. Benedetto and S. Li, “The theorem of multiresolution analysis frames and application to filter banks,” Appl. Comput. Harmon. Anal., vol. 5, 1998, pp. 398-427.
[11]	C. K. Chui and W. He, “Compactly supported tight frames associated with refinable functions,” Appl. Comp. Harmonic Anal., vol. 8, 2000, pp. 293-319.
[12]	W. Lawton, S. L. Lee and Z. Shen, “An algorithm for matrix extension and wavelet construction,” Math. Comp., vol. 65, 1996, pp. 723-737.
[13]	Cui Lihong, Cheng Zhengxing and Yang Shouzhi, “Explicit Structuer of Wavelet Tight Frames,” Acta Mathematica Scientia, vol. 24 (1), 2004, pp. 94-104.
[14]	Sun Qiyu, Bi Ning and Huang Daren, An introduction to multiband wavelets, Zhejiang university press, 2001.

Cite This Article

Plain Text BibTeX RIS

APA Style

Yuanyuan Zhang, Zhaofeng Li. (2018). The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA. Applied and Computational Mathematics, 7(3), 161-166. https://doi.org/10.11648/j.acm.20180703.22

Copy | Download

ACS Style

Yuanyuan Zhang; Zhaofeng Li. The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA. Appl. Comput. Math. 2018, 7(3), 161-166. doi: 10.11648/j.acm.20180703.22

Copy | Download

AMA Style

Yuanyuan Zhang, Zhaofeng Li. The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA. Appl Comput Math. 2018;7(3):161-166. doi: 10.11648/j.acm.20180703.22

Copy | Download

@article{10.11648/j.acm.20180703.22,
  author = {Yuanyuan Zhang and Zhaofeng Li},
  title = {The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA},
  journal = {Applied and Computational Mathematics},
  volume = {7},
  number = {3},
  pages = {161-166},
  doi = {10.11648/j.acm.20180703.22},
  url = {https://doi.org/10.11648/j.acm.20180703.22},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180703.22},
  abstract = {In 1974, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing. In 1986, Y. Meryer created a real small wave base, and the wavelet analysis began to flourish after a multi scale analysis of the same method of constructing the small wave base with S. Mallat. In order to analyze and deal with non-stationary signals, a series of new signal analysis theories are proposed.: Short Time Fourier Transform, time-frequency analysis, wavelet transform, and fractional Fourier transform and so on. In this paper, an explicit algorithm is given to construct the minimum-energy frames based on frame multiresolution analysis via characteristic vectors of the mask matrix. In section 2, we show the structure of minimum-energy wavelet frames in terms of their masks (Lemma 1) and discuss that we should eliminate the correlation of the rows of the mask matrix by the polyphase decomposition technique. Based on FMRA, an explicit algorithm is given to construct this frames. By this method, all the minimum-energy wavelet frames can be obtained. As an application, several examples are showed to explain this method in section 3. This method can also be applied in other fields of wavelet analysis.},
 year = {2018}
}

Copy | Download

TY - JOUR
T1 - The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA
AU - Yuanyuan Zhang
AU - Zhaofeng Li
Y1 - 2018/08/13
PY - 2018
N1 - https://doi.org/10.11648/j.acm.20180703.22
DO - 10.11648/j.acm.20180703.22
T2 - Applied and Computational Mathematics
JF - Applied and Computational Mathematics
JO - Applied and Computational Mathematics
SP - 161
EP - 166
PB - Science Publishing Group
SN - 2328-5613
UR - https://doi.org/10.11648/j.acm.20180703.22
AB - In 1974, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing. In 1986, Y. Meryer created a real small wave base, and the wavelet analysis began to flourish after a multi scale analysis of the same method of constructing the small wave base with S. Mallat. In order to analyze and deal with non-stationary signals, a series of new signal analysis theories are proposed.: Short Time Fourier Transform, time-frequency analysis, wavelet transform, and fractional Fourier transform and so on. In this paper, an explicit algorithm is given to construct the minimum-energy frames based on frame multiresolution analysis via characteristic vectors of the mask matrix. In section 2, we show the structure of minimum-energy wavelet frames in terms of their masks (Lemma 1) and discuss that we should eliminate the correlation of the rows of the mask matrix by the polyphase decomposition technique. Based on FMRA, an explicit algorithm is given to construct this frames. By this method, all the minimum-energy wavelet frames can be obtained. As an application, several examples are showed to explain this method in section 3. This method can also be applied in other fields of wavelet analysis.
VL - 7
IS - 3
ER -

Copy | Download

Author Information

Yuanyuan Zhang

College of Sciences, China Three Gorges University, Yichang, China
Zhaofeng Li

College of Sciences, China Three Gorges University, Yichang, China

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Yuanyuan Zhang, Zhaofeng Li. (2018). The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA. Applied and Computational Mathematics, 7(3), 161-166. https://doi.org/10.11648/j.acm.20180703.22

Copy | Download

ACS Style

Yuanyuan Zhang; Zhaofeng Li. The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA. Appl. Comput. Math. 2018, 7(3), 161-166. doi: 10.11648/j.acm.20180703.22

Copy | Download

AMA Style

Yuanyuan Zhang, Zhaofeng Li. The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA. Appl Comput Math. 2018;7(3):161-166. doi: 10.11648/j.acm.20180703.22

Copy | Download

@article{10.11648/j.acm.20180703.22,
  author = {Yuanyuan Zhang and Zhaofeng Li},
  title = {The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA},
  journal = {Applied and Computational Mathematics},
  volume = {7},
  number = {3},
  pages = {161-166},
  doi = {10.11648/j.acm.20180703.22},
  url = {https://doi.org/10.11648/j.acm.20180703.22},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180703.22},
  abstract = {In 1974, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing. In 1986, Y. Meryer created a real small wave base, and the wavelet analysis began to flourish after a multi scale analysis of the same method of constructing the small wave base with S. Mallat. In order to analyze and deal with non-stationary signals, a series of new signal analysis theories are proposed.: Short Time Fourier Transform, time-frequency analysis, wavelet transform, and fractional Fourier transform and so on. In this paper, an explicit algorithm is given to construct the minimum-energy frames based on frame multiresolution analysis via characteristic vectors of the mask matrix. In section 2, we show the structure of minimum-energy wavelet frames in terms of their masks (Lemma 1) and discuss that we should eliminate the correlation of the rows of the mask matrix by the polyphase decomposition technique. Based on FMRA, an explicit algorithm is given to construct this frames. By this method, all the minimum-energy wavelet frames can be obtained. As an application, several examples are showed to explain this method in section 3. This method can also be applied in other fields of wavelet analysis.},
 year = {2018}
}

Copy | Download